CAMP Special Lecture
This series of lectures is free and open to everyone.
If you would join this, in order to let us know, please register here before June 14.
- Title. Alternating Permutations
- Lecturer. Richard P. Stanley, MIT, USA
- Date. June 29 (Mon.) - July 3 (Fri.), 2015
- Time. 14:00 (June 29-30, July 1, July 3), 15:30 (July 2)
- Venue. CAMP & NIMS
- Detailed Time & Venue.
- NIMS Seminar Room (3rd floor), 2:00 PM, June 29, 2015
- NIMS Seminar Room (3rd floor), 2:00 PM, June 30, 2015
- CAMP Seminar Room (M), 2:00 PM, July 1, 2015
- CAMP Seminar Room (L), 3:30 PM, July 2, 2015
- CAMP Seminar Room (L), 2:00 PM, July 3, 2015
- Abstract. (for all lectures) A permutation a1 a2 ... an of 1, 2, ..., n is alternating if a1 > a2 < a3 > a4 < ... . Alternating permutations have many fascinating properties related to combinatorics, geometry, representation theory, probability theory, and other areas. We will survey the highlights of this subject.
Lecture 1. (June 29, 14:00) Euler numbers and André's theorem
The number of alternating permutations of 1, 2, ..., n is denoted En and is called an Euler number. The first theorem on alternating permutations is the generating function
∑n≥0 En xn / n! = tan(x) + sec(x) due to D. André in 1879. Some other combinatorial objects are also counted by the Euler numbers.
Lecture 2. (June 30, 14:00) Convex polytopes and alternating permutations
There are two interesting convex polytopes whose volume is En / n!. One of them is given by xi ≥ 0 (1 ≤ i ≤ n) and xi + xi+1 ≤ 1 (1 ≤ i ≤ n-1). They fit into a more general context of order polytopes and chain polytopes of posets.
Lecture 3. (July 1, 14:00) Longest alternating subsequences
What can one say about the length of the longest alternating subsequence of a random permutation? This theory is analogous but simpler than the better-known theory of the longest increasing subsequence of a random permutation. For example, if n>1 then the expected length of the longest alternating subsequence of a random permutation of 1, 2, ..., n is exactly (4n+1)/6.
Lecture 4. (July 2, 15:30) A symmetric group character related to alternating permutations
H. O. Foulkes defined a certain character χ of the symmetric group Sn whose values χ(w) are either 0 or ±Ek for some 1 ≤ k ≤ n (depending on w). This character allows us to enumerate certain classes of alternating permutations using representation theory. For instance, if p is prime then the number of alternating permutations in Sp that are also p-cycles is equal to
(Ep - (-1)(p-1)/2) / p. No elementary proof is known.
Lecture 5. (July 3, 14:00) The cd-index of the symmetric group
The cd-index of Sn is a noncommutative polynomial in the indeterminates c and d that encodes a lot of combinatorial information. The number of terms in this polynomial is En, and there are interesting connections with alternating permutations.
|Byunghak Hwang ||Seoul National University ||Student |
|Heesung Shin ||Inha University ||Professor |
|James Haglund ||University of Pennsylvania, USA ||Professor |
|Jiang Zeng ||Université Claude Bernard Lyon 1, France ||Professor |
|Jinha Kim ||Seoul National University ||Student |
|Jung Seok Oh ||Seoul National University ||Student |
|Kang-Ju Lee ||Seoul National University ||Student |
|Kyoungsuk Park ||Ajou University ||Student |
|Masao Ishikawa ||University of the Ryukyus, Japan ||Professor |
|Minki Kim ||KAIST ||Student |
|Richard P. Stanley ||MIT, USA ||Professor |
|Sang-Hoon Yu ||Seoul National University ||Student |
|Sangwook Kim ||Chonnam National University ||Professor |
|Sanha Lee ||Sungkyunkwan University ||Student |
|Seung-Il Choi ||Sogang University ||Postdoc |
|Seung Jin Lee ||KIAS ||Postdoc |
|Shishuo Fu ||Chongqing University, China ||Professor |
|Sunyoung Nam ||Sogang University ||Student |
|YoungHun Kim ||Sogang University ||Student |
|Younjin Kim ||KAIST ||Postdoc |
|Zhicong Lin ||NIMS ||Postdoc |
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